Finite Group Databasebeta

← C2×#11751606 #14513237

Subgroups — C2×#11751606 #14513237

Order 1920.

698 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2C22×C5×#226C211yesmaximal, characteristic, complemented
3C2×#91184C211yesmaximal, characteristic, complemented
4#11751606C214yesmaximal, direct product factor
5C2×#65475—31nomaximal, Hall
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×D5×SL(2,𝔽3)—22no—
8C2×D5×SL(2,𝔽3)C2212yes—
9C2×C5×#226—22no—
10C2×C5×#226C2212yescomplemented
11C2×#6665—41nomaximal
12#91184C2212yescomplemented
13C2×#5419—51nomaximal, Hall
14C22×Q8×D5D311yescharacteristic
15C22×C5×Q16—31no—
16C2×#4092—31no—
17C22×#689—31no—
18C2×#3967—31no—
19#65475—34no—
20#65475—34no—
21C2×#3952—31no—
22C2×#3948—31no—
23C23×C3×D5—41no—
24C22×Dic15—41no—
25C22×C5×Dic3—41no—
26C2×C5×SL(2,𝔽3)D412yes—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28D5×SL(2,𝔽3)—22no—
29C5×#226—22no—
30#6665—44no—
31C23×SL(2,𝔽3)—51no—
32C22×#226D511yescharacteristic, complemented
33#5419—54no—
34C2×#859—51no—
35C22×C5×Q8—31no—
36C22×C5×Q8C2×D311yescharacteristic
37C2×Q8×D5—22no—
38C2×Q8×D5—32no—
39C2×Q8×D5—62no—
40C2×Q8×D5C2×D312yes—
41C22×C4×D5—31no—
42C22×Dic10—31no—
43C2×C5×Q16—32no—
44C2×C5×Q16—32no—
45C2×C5×Q16—62no—
46C2×C5×Q16—62no—
47C22×C5×C8—31no—
48#4092—34no—
49#4092—62no—
50C2×#689—34no—
51C2×#689—64no—
52C2×#685—61no—
53C2×C4⋊Dic5—31no—
54C2×#683—61no—
55C22×#168—31no—
56#3967—32no—
57#3967—32no—
58#3952—34no—
59#3948—34no—
60C2×#451—151noSylow
61C23×C3×C5—41no—
62C22×C3×D5—44no—
63C22×C3×D5—42no—
64C22×C3×D5—82no—
65C22×C3×D5—82no—
66C2×Dic15—42no—
67C2×Dic15—82no—
68C2×C5×Dic3—42no—
69C2×C5×Dic3—82no—
70C5×SL(2,𝔽3)C2×D411yescharacteristic
71C22×SL(2,𝔽3)—52no—
72C22×SL(2,𝔽3)—102no—
73C22×SL(2,𝔽3)C2×D511yescharacteristic
74C2×#226—22no—
75C2×#226C2×D512yescomplemented
76C2×#217—201no—
77#859—52no—
78C23×D5S411yescharacteristic
79C2×C5×Q8—31no—
80C2×C5×Q8—32no—
81C2×C5×Q8—32no—
82C2×C5×Q8—61no—
83C2×C5×Q8—62no—
84C2×C5×Q8—62no—
85C2×C5×Q8#6712yes—
86C2×C5×Q8C22×D311yescharacteristic
87C22×C4×C5—31no—
88C22×C4×C5—61no—
89C22×Dic5—31no—
90C22×Dic5—61no—
91Q8×D5—22no—
92Q8×D5—62no—
93Q8×D5—64no—
94Q8×D5—64no—
95C2×C4×D5—32no—
96C2×C4×D5—32no—
97C2×C4×D5—62no—
98C2×C4×D5—62no—
99C2×Dic10—32no—
100C2×Dic10—34no—
101C2×Dic10—61no—
102C2×Dic10—62no—
103C5×Q16—64no—
104C5×Q16—62no—
105C5×Q16—62no—
106C2×C5×C8—32no—
107C2×C5×C8—62no—
108#689—68no—
109#685—64no—
110C4⋊Dic5—32no—
111C4⋊Dic5—61no—
112#683—64no—
113C2×#168—32no—
114C2×#168—62no—
115C23×Q8—51no—
116C22×Q16—31no—
117C22×Q16—151no—
118C2×C22⋊Q8—151no—
119#451—154no—
120#451—154no—
121C2×#103—151no—
122C2×#103—151no—
123C2×C22⋊C8—151no—
124C22×C3×C5—42no—
125C22×C3×C5—42no—
126C22×C3×C5—42no—
127C22×C3×C5—41no—
128C2×C3×D5—44no—
129C2×C3×D5—82no—
130C2×C3×D5—82no—
131C2×C3×D5—88no—
132Dic15—82no—
133C5×Dic3—82no—
134C24×C3—201no—
135C22×Dic3—41no—
136C22×Dic3—201no—
137C2×SL(2,𝔽3)—102no—
138C2×SL(2,𝔽3)C22⋊D512yes—
139C2×SL(2,𝔽3)C22×D511yescharacteristic
140#226—22no—
141#217—204no—
142C23×C5C2×S411yescharacteristic
143C22×D5—22no—
144C22×D5—22no—
145C22×D5#22614yescomplemented
146C22×D5C2×S412yes—
147C5×Q8—32no—
148C5×Q8—31no—
149C5×Q8—61no—
150C5×Q8—62no—
151C5×Q8—62no—
152C5×Q8—122no—
153C5×Q8C2×#6711yescharacteristic
154C2×C4×C5—32no—
155C2×C4×C5—32no—
156C2×C4×C5—31no—
157C2×C4×C5—31no—
158C2×C4×C5—62no—
159C2×C4×C5—122no—
160C2×Dic5—32no—
161C2×Dic5—62no—
162C2×Dic5—62no—
163C2×Dic5—122no—
164C4×D5—62no—
165C4×D5—62no—
166C4×D5—64no—
167Dic10—34no—
168Dic10—64no—
169Dic10—62no—
170C5×C8—62no—
171#168—62no—
172C22×Q8—31no—
173C22×Q8—52no—
174C22×Q8—102no—
175C22×Q8—151no—
176C22×Q8—152no—
177C22×Q8—302no—
178C22×Q8D3×D511yescharacteristic
179C23×C4—151no—
180C2×Q16—32no—
181C2×Q16—32no—
182C2×Q16—62no—
183C2×Q16—62no—
184C2×Q16—154no—
185C2×Q16—304no—
186C22×C8—31no—
187C22×C8—151no—
188C22⋊Q8—154no—
189C22⋊Q8—302no—
190C2×C4⋊C4—151no—
191C2×C4⋊C4—301no—
192C2×C22⋊C4—301no—
193#103—154no—
194#103—152no—
195#103—152no—
196C22⋊C8—154no—
197C2×C3×C5—41no—
198C2×C3×C5—41no—
199C2×C3×C5—41no—
200C2×C3×C5—44no—
201C3×D5—84no—
202C23×C3—41no—
203C23×C3—202no—
204C23×C3—204no—
205C23×C3—402no—
206C23×C3—402no—
207C2×Dic3—42no—
208C2×Dic3—82no—
209C2×Dic3—202no—
210C2×Dic3—402no—
211SL(2,𝔽3)C2×C22⋊D511yescharacteristic
212C22×C5C2×#22612yes—
213C22×C5#98312yes—
214C22×C5GL(2,ℤ/4)12yes—
215C22×C5C22×S411yescharacteristic
216C2×D5—22no—
217C2×D5—22no—
218C2×D5—28no—
219C2×D5C2×#22614yescomplemented
220C4×C5—31no—
221C4×C5—31no—
222C4×C5—32no—
223C4×C5—122no—
224Dic5—62no—
225Dic5—122no—
226C24—51no—
227C2×Q8—31no—
228C2×Q8—32no—
229C2×Q8—32no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—62no—
233C2×Q8—102no—
234C2×Q8—152no—
235C2×Q8—154no—
236C2×Q8—301no—
237C2×Q8—302no—
238C2×Q8—304no—
239C2×Q8—302no—
240C2×Q8—304no—
241C2×Q8#120112yes—
242C2×Q8C2×D3×D511yescharacteristic
243C22×C4—31no—
244C22×C4—61no—
245C22×C4—152no—
246C22×C4—152no—
247C22×C4—151no—
248C22×C4—302no—
249C22×C4—302no—
250C22×C4—301no—
251Q16—64no—
252Q16—62no—
253Q16—62no—
254Q16—308no—
255C2×C8—32no—
256C2×C8—62no—
257C2×C8—152no—
258C2×C8—302no—
259C4⋊C4—152no—
260C4⋊C4—301no—
261C4⋊C4—304no—
262C22⋊C4—304no—
263C3×C5—41noHall
264C22×C3—42no—
265C22×C3—41no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—204no—
269C22×C3—408no—
270C22×C3—402no—
271C22×C3—402no—
272Dic3—82no—
273Dic3—402no—
274C2×C5#541914yescomplemented
275C2×C5C22×#22611yescharacteristic
276C2×C5C2×#98311yescharacteristic
277C2×C5C2×GL(2,ℤ/4)11yescharacteristic
278D5—24no—
279C23—52no—
280C23—54no—
281C23—102no—
282C23—102no—
283C23D5×S411yescharacteristic
284Q8—32no—
285Q8—31no—
286Q8—61no—
287Q8—62no—
288Q8—62no—
289Q8—122no—
290Q8—154no—
291Q8—302no—
292Q8—304no—
293Q8C2×#120111yescharacteristic
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—62no—
299C2×C4—122no—
300C2×C4—152no—
301C2×C4—302no—
302C2×C4—302no—
303C2×C4—304no—
304C2×C4—302no—
305C2×C4—302no—
306C2×C4—602no—
307C8—62no—
308C8—302no—
309C2×C3—44no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—404no—
314C5C2×#541911yescharacteristic, complemented, Sylow
315C22—54no—
316C22—108no—
317C22—102no—
318C22—102no—
319C22D5×#22612yes—
320C22#9189712yes—
321C22#9190412yes—
322C22C2×D5×S411yescharacteristic
323C4—31no—
324C4—31no—
325C4—32no—
326C4—122no—
327C4—302no—
328C4—602no—
329C3—41noSylow
330C2—104no—
331C2#1175160614yesdirect product factor
332C2C2×D5×#22611yescharacteristic
333C2C2×#9189711yescharacteristic
334C2C2×#9190411yescharacteristic
subgroupG/H
center, hypercenterC23D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FrattiniC22C2×D5×S4
FittingC22×C5×Q8C2×D3
socleC23×C5C2×S4
prime radicalC22×D5×SL(2,𝔽3)C2
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×C5×#226C2semidirect
3. C2×#91184C2semidirect
4. #11751606C2direct (N × H)
10. C2×C5×#226C22semidirect
12. #91184C22semidirect
32. C22×#226D5semidirect
75. C2×#226C2×D5semidirect
145. C22×D5#226semidirect
219. C2×D5C2×#226semidirect
274. C2×C5#5419semidirect
314. C5C2×#5419semidirect
331. C2#11751606direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×D5×SL(2,𝔽3)2
C22×C5×#2262
C2×#911842
#117516062
C2×#654753
C2×#66654
C2×#54195